Jordan Turner-Hall of Harlequins is tackled by Niall O'Connor, left, of Connacht during the Heineken Cup match in Galway. Photograph: Ian Walton/Getty Images

Connacht's first victory in four months left Harlequins on the verge of being eliminated from the Heineken Cup. The Premiership leaders need results in two groups this weekend to go their way to qualify for the last eight as one of the best runners-up after failing to capitalise on Toulouse's defeat at Gloucester and blowing the use of a gale in the second half when they scored a mere three points.

Quins opted to play into the strong wind in the first half, keeping hold of the ball rather than kicking, but the ploy became increasingly hazardous with the rain prompting a number of handling errors.

Connacht were an attacking threat only towards the end of the first period, but they were handed six points in the opening 12 minutes when James Johnston twice slipped his binding in a dominant scrum and Niall O'Connor mastered the swirling wind to kick two penalties.

In between, Sam Smith scored a smart try for Quins. The Premiership leaders took play through eight phases before Nick Evans looped around Jordan Turner-Hall, and although the fly-half's long pass forced Smith to take the ball standing still, the wing stepped away from Fetu'u Vainikolo and touched the ball down an instant before being tackled into touch in goal.

Quins were dominating possession, but they wasted a scrum five when an Evans fumble allowed Connacht to hoof the ball upfield. Danny Care was also hustled out of possession and Connacht, who had lost their previous 14 matches since beating Newport Gwent Dragons back in September, tried to make play scrappy.

The home side led 9-5 at the interval when O'Connor converted Tomas Vallejos's indiscretion at a breakdown into three points but the fly-half missed his least difficult penalty of the half shortly before the break after Chris Robshaw played the ball in an offside position near his own line.

Quins used the wind to peg back Connacht and opted for scrums after being awarded two kickable penalties on 49 minutes. The ploy came to nothing but for all the turnovers the home side forced, they could not get out of their own half with relieving kicks held up in the wind. It took Connacht 17 minutes of the second half to get into Quins' territory. Luke Wallace, who had come on for the injured Maurie Fa'asavalu, was penalised for holding on, but the raid lasted seconds before the ball was turned over and kicked downfield.

Evans kicked a penalty on the hour after the outstanding John Muldoon was caught offside, but Quins missed the chance to regain the lead when Evans hooked another penalty from a scrum and they were nine minutes away from a shock defeat.

"It's hard to put into words how disappointed we are, it's a very low point," admitted captain Chris Robshaw. "We're going to have to console ourselves tonight, look back on the video and hope that someone else slips up along the line. No one was complacent. We wanted to be in the knockout stages. Obviously we'll see where things end up on Sunday night, but in all honesty we probably will be back in the Amlin [defending the title we won last year]."